Certainty & Uncertainty: Logical Probability & Statistics

Since we have spent the weekend with these matters, I thought it appropriate to include the first part of the Introduction to the new book I’ve been working on. It is only slightly similar to the old book.

There are things we know with certainty. These things are true or false given some premises or evidence or just because upon reflection they are obviously true or false. There are many more things about which we are uncertain. These things too are more or less probable given certain premises and evidence. And there are still more things of which nobody can ever specify the uncertainty. These things are nonsensical or paradoxical.

The truth, falsity, or in-betweenness of any proposition can only be known with respect to stated evidence or premises. We know that given the premises “All men are mortal” and “Socrates is a man” that the proposition “Socrates is mortal” is true. Given other premises the same proposition may be true, false, or in-between, which is to say merely probable. Swapping the first premise with “No men are mortal” changes the truth of the final proposition to falsity. Exchange it with “Most men are mortal” and the truth or falsity of the final proposition can no longer be ascertained, though its probability can. The probability of “Socrates is mortal” given “All men are mortal and Socrates is a man” is 1; just as the probability of the same proposition given “No men are mortal, etc.” is 0. And the probability of the same proposition given “Most men, etc.” is greater than 0.5 but less than 1. This, incidentally, shows that probability is often an interval. This result only follows because we tacitly include a premise about the definition of the English word most; here its definition means “a majority but not all.”

This move is perfectly acceptable, even if unfamiliar. Consider you are supplying the argument with many tacit assumptions, such as definitions for all, men, and so forth, along with premises about how the words All men, etc. in sequence are turned into English with a definite meaning, about how men are discrete individuals, and so on. Thus if you quibble with my definition of most, you are free to substitute your own, as long as you make it clear just what definition you hold. As we shall see, debates about the probability of a proposition are really about the list and meanings of premises.

Change the first premise to “All men are moral” (notice the absence of t) and then nothing can be said about the proposition “Socrates is mortal.” A man’s morality has no bearing on his mortality (though it might affect his immortality). The proposition given these premises is clearly not true, and just as clearly not false. It also has no probability because there is no evidence in the premises which are probative to the proposition before us. The probability the proposition is true given these premises is undefined.

This also should not be strange. Consider any proposition you like, such as “Jack can lift 100 pounds.” What is the probability it is true? There is none because no premises have been supplied. Suppose I offer as a premise “2 < 4” and re-ask the question. Still no answer because this mathematical fact has no bearing on the proposition.

Or perhaps this tacit premise has suggested itself, “Jack can either lift 100 pounds or he cannot”; therefore, given this premise the probability the proposition is true is 0.5. This is false. It is a deducible truth of logic that adding any truth to a list of premises, or to a proposition does not change the logical status of that proposition. So if we prefix “No men, etc.” with “T & No men, etc.” the proposition “S. is mortal” given these premises remains false, and where T is any truth. The premise “Jack can either lift 100 pounds or he cannot” is a truth; it is a tautology and tautologies are always true. It is just as true as “Jack either has cancer or he doesn’t.” That being so, this latter truth can be substituted with the first truth, where it is now obviously unrelated to the proposition.

One other tacit premise lurks: this is that the proposition about Jack’s muscle power suggests contingency. Contingent propositions are events which we know, via a multitude of paths, are not necessary truths or falsities. This “premise” is actually a host of premises which lead us to conclude that we cannot (it is impossible to) find a formal proof that makes the proposition about Jack a necessary truth or falsity (this sentence is the conclusion to that argument). But accepting this premise, or premises, does not buy us much, for given this premise, or premises, the probability of the proposition is greater than 0 and less than 1, and that is the best we can do. This merely says, in quantifiable terms, that the proposition is not a truth or falsity.

Logic, of which probability is a branch, is concerned only with the connections between premises and conclusions and not with the premises or conclusions themselves; and this is so whether we discuss their veracity or origin. Thus when we say, given “All men, etc.” the probability that “S. is mortal” is 1, we are not casting judgment on the premise “All men are mortal” nor are we concerned (at this point) where the “conclusion” “Socrates is mortal” arose. The premise “All men are mortal” indeed appears false—but only because we implicitly add premises such as Benjamin Franklin’s which encapsulate all human experience about man’s limited stay on the planet. Since these premises did not appear in the original argument, we are not free to put them there, at least not when considering the argument as it stands, or for “argument’s sake”, or when demonstrating logical principles. The proposition which is the conclusion was also supplied to us, and we must be ever careful to keep it and not exchange it for another; at least not without being clear that a modification is being made.

From this we can conclude that Dr. Dodgson was right when he wrote that given, “All cats are creatures understanding French” and “Some chickens are cats” that the proposition “Some chickens are creatures understanding French” is true—and deduced to be true at that. But then all probabilities, just as all statements of logic, are deduced. It makes not a whit of difference that given the premise “Nobody ever observed a cat understanding French” Dodgson’s first premise was false. What matters is that the argument as it stands leads to a valid conclusion, that we deduce the probability of its conclusion (relative to the premises) as 1. And thus from the premises “Half of all Martians wear hats” and “George is a Martian” we judge the probability of the proposition “George wears a hat” to be 0.5, and no other number. Note that the first premise contains the tacit premise that the number of Martians is divisible by 2, unless we allow the colloquialism that half means “about half.” If so, then the probability of the conclusion is about 0.5. The quantification cannot be made more precise than the language.